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Answering a longstanding problem originating in Christensen’s seminal work on Haar null sets [ Math. Scand.   28 (1971), 124–128; Israel J. Math.   13 (1972), 255–260; Topology and Borel Structure. Descriptive Topology and Set Theory with Applications to Functional Analysis and Measure Theory , North-Holland Mathematics Studies, 10 (Notas de Matematica, No. 51). (North-Holland Publishing Co., Amsterdam–London; American Elsevier Publishing Co., Inc., New York, 1974), iii+133 pp], we show that a universally measurable homomorphism between Polish groups is automatically continuous. Using our general analysis of continuity of group homomorphisms, this result is used to calibrate the strength of the existence of a discontinuous homomorphism between Polish groups. In particular, it is shown that, modulo $$\text{ZF}+\text{DC}$$ , the existence of a discontinuous homomorphism between Polish groups implies that the Hamming graph on $$\{0,1\}^{\mathbb{N}}$$ has finite chromatic number.